Add a little vinegar to the salted water. It will keep the white from running in the event there is a crack while boiling. Fresh eggs are harder to peel because they have more shell membrane that cling tenaciously to the shell. As they get older, the membrane begins to shrink and leave more of an air pocket.
Start peeling at the larger end, where the air pocket is, and remove the shell under running water to make the shelling easier. You have get a hold of the membrane under the shell when you remove the it.